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5. Image Shading
In paragraph 2.2 and fig. 2.2 we defined the field angle. Please look at this figure again.

Imagine that your eye is located in the object center point and you look at the lens.
Now you move your eye upwards to the top object corner   --   always staring at the lens.
What you see is the front lens element with decreasing area, see fig. 5a.

fig.5a: illuminance decay (14kByte)

The "apparent" (that is "efficient") area decreases with the cosine of the halved field angle.

The radiating object rim area that the lens sees, too, is shrinked by the cosine of the halved field angle.   --   Makes cosine squared.

Now we move to image space, where the same applies:

When the exit pupil illuminates the image rim, transmitting and receiving area each are apparently shrinked by the cosine of the halved image angle.   --   Makes cosine to the fourth.

So we understand why "natural" shading of an image is governed by the cosine-to-the-fourth-law.

In fact real lens systems exhibit even more shading:

Relatively complex is the evaluation of those laws that govern the angle-dependent reflection on glass surfaces.
But you surely feel that the more you go from normal incidence to grazing incidence, the more reflection will occur. This is one of the reasons why the rays at the rim of the lens aperture are more difficult to control than in the lens center.
And why at large field angles, image illumination gets worse.

But with simple means of geometrical optics you can understand the effect of "vignetting":

fig.5b: Vignetting (8kByte)

Fig. 5b on its left side shows an eye. This eye should be focused for infinity or illuminated from the right side with a telecentric (i.e. parallel) bundle.

Arranged like in Fig. 5b top, the entire aperture of the cylindrical tube allows the light to pass through to the eye.

But arranged like in Fig. 5b bottom, we see that the effective aperture is greatly reduced by slanting the tube.

From this we learn: If in an objective lens system   the free apertures of all lens elements are used up entirely,   then vignetting will govern the image light distribution.

Things improve if you close the f-stop a bit more. This way, incidance distribution is flattened; but always you can only level down.

For calculating the degree of vignetting you would need several internal lens properties. Pretty difficult to get. Better do some tests and measurements. Or insist on getting a complete lens data sheet.

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Last modified Nov.28th, 2002 23:32